The 21-year-old was in Naples at the end of June to tour the city, as talks continued between the clubs over a €50m-plus deal.
There have been reports that he would prefer a Premier League move and is waiting for offers from England.
However, his brother-in-law Osita Okolo, a member of Osimhen’s camp, said Victor had enjoyed Naples.
“Victor will make a choice in the next few hours,” Okolo told TMW. “The city, the people and Napoli have made a good impression on him.
“Victor met [Aurelio] De Laurentiis and [Rino] Gattuso and felt he would be an important player in the blue.
“But now we are waiting for his decision. Osimhen is reflecting and will make his announcement in the next few hours.”
The Nigeria forward scored 18 goals in 38 appearances across all competitions for LOSC this season.